where we use the definition \(z\equiv m_\chi /T\) , and \(T\)  is the temperature. The parameters \(k\)  and \(k^{\star }\)  are denoted as \(k=\sqrt{\pi g_\star /45}m_\chi M_{Pl}\)  and \(k^{*}=\sqrt{45/4\pi ^{3}g_{\star }}M_{Pl}/m_\chi ^2\)  respectively, where \(g_{\star }\)  is the effective number of degrees of freedom of the relativistic species and \(M_{Pl}=1.2 \times 10^{19}\)  GeV is the Planck mass. In the above Boltzmann equations, contributions from scattering processes as \(\phi \phi \rightarrow \chi \chi \) , \(NN\rightarrow \chi \chi \) , \(NN\rightarrow \phi \phi \) , \(h\nu \rightarrow \phi \chi \)  and \(hN\rightarrow \phi \chi \)  are not considered, since the corresponding cross sections are suppressed [1]}, [2]}. We use micrOMEGAs [3]} to calculate the thermal average cross sections \(\langle \sigma v \rangle \) . The thermal decay width \(\tilde{\Gamma }_i\)  is defined as \(\Gamma _i \mathcal {K}_1/\mathcal {K}_2\)  with \(\mathcal {K}_{1,2}\)  being the first and second modified Bessel Function of the second kind.
Corresponding decay widths are given by
\(\Gamma _{{N\rightarrow }{\phi \chi }}&=&\frac{\lambda _{ds}^{2}}{16\pi }\frac{(m_N+m_\chi )^2-m_\phi ^2}{m_N^3}{\lambda ^{1/2}(m_N^2,m_\phi ^2,m_\chi ^2)}, \\\Gamma _{\phi \rightarrow \chi \nu } &=& \frac{\lambda _{ds}^{2}\theta ^2\, m_\phi }{8\pi }{\left(\frac{m_\phi ^2-m_\chi ^2}{m_\phi ^2}\right)^2}.\) 
